Testing for Symmetry and Conditional Symmetry Using Asymmetric Kernels

We derive nonparametric tests of symmetry using asymmetric kernels with either shrinking or fixed bandwidths. We show how to extend the approach to examine conditional symmetry by deriving conditions under which our tests are applicable to residuals from semiparametric models with a (sufficiently smooth) nonparametric link function. As a by-product, we prove the consistency of the asymmetric kernel estimator of the derivative of the density function. Simulations show that the asymptotic tests perform well even in very small samples, entailing better size and power properties than some of the existing symmetry tests.